TAILIEUCHUNG - Báo cáo toán học: " et another distance-regular graph related to a Golay code"

YTuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: et another distance-regular graph related to a Golay code. | Yet another distance-regular graph related to a Golay code Leonard H. Soicher School of Mathematical Sciences Queen Mary and Westfield College Mile End Road London E1 4NS . email Submitted January 20 1995 Accepted January 22 1995 Abstract We describe a new distance-regular but not distance-transitive graph. This graph has intersection array 110 81 12 1 18 90 and automorphism group M22 2. In 1 Brouwer Cohen and Neumaier discuss many distance-regular graphs related to the famous Golay codes. In this note we describe yet another such graph. Ivanov Linton Lux Saxl and the author 4 have classified all primitive distance-transitive representations of the sporadic simple groups and their automorphism groups. As part of this work all multiplicity-free primitive representations of such groups have also been classified. One such representation is M22 2 on the cosets of L2 11 2. This representation has rank 6 with subdegrees 1 55 55 66 165 330. Let r be the graph obtained by the edge-union of the orbital graphs corresponding to the two suborbits of length 1991 Mathematics Subject Classification 05E30 05C25 1 THE ELECTRONIC .JOURNAL OF COMBINATORICS 2 1995 N1 2 55. By examining the sum 0 55 0 0 0 0 0 0 55 0 0 0 1 8 4 0 18 24 0 4 12 0 3 36 0 4 12 0 3 36 1 1 12 0 0 30 12 0 0 0 10 10 35 0 0 0 10 20 25 0 6 1 4 20 24 0 1 10 8 4 32 0 4 6 7 12 26 J 0 6 2 5 16 26 of the intersection matrices corresponding to these two orbital graphs we see that r is distance-regular with intersection array 110 81 12 1 18 90 . According to 1 this graph was previously unknown. We now give a description of r in terms of a punctured binary Golay code. This description was obtained using the GRAPE share library package 7 of the GAP system 6 available from . Let C22 be the code obtained by puncturing in one co-ordinate the nonextended binary Golay code. Then C22 is a 22 12 6 -code with automorphism group M22 2. Let M be the set of the 77 minimum weight

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